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Theorem opid 3638
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
Hypothesis
Ref Expression
opid.1 𝐴 ∈ V
Assertion
Ref Expression
opid 𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid
StepHypRef Expression
1 dfsn2 3458 . . . 4 {𝐴} = {𝐴, 𝐴}
21eqcomi 2092 . . 3 {𝐴, 𝐴} = {𝐴}
32preq2i 3521 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4 opid.1 . . 3 𝐴 ∈ V
54, 4dfop 3619 . 2 𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6 dfsn2 3458 . 2 {{𝐴}} = {{𝐴}, {𝐴}}
73, 5, 63eqtr4i 2118 1 𝐴, 𝐴⟩ = {{𝐴}}
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  Vcvv 2619  {csn 3444  {cpr 3445  cop 3447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453
This theorem is referenced by:  dmsnsnsng  4903  funopg  5042
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